A241701 - OEIS

%I #21 Apr 30 2017 09:49:38

%S 1,1,1,2,1,2,2,3,4,4,8,2,5,13,5,6,21,12,8,33,27,3,10,50,53,11,12,73,

%T 98,31,15,106,174,78,5,18,150,296,175,22,22,209,486,363,72,27,289,781,

%U 715,204,8,32,393,1222,1342,510,43,38,529,1874,2421,1168,159

%N Number T(n,k) of Carlitz compositions of n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

%C No two adjacent parts of a Carlitz composition are equal.

%H Alois P. Heinz, <a href="/A241701/b241701.txt">Rows n = 0..250, flattened</a>

%F Sum_{k=0..floor(n/3)} (k+1) * T(n,k) = A285994(n) (for n>0).

%e T(6,0) = 4: [6], [1,5], [2,4], [1,2,3].

%e T(6,1) = 8: [4,2], [5,1], [3,1,2], [1,3,2], [1,4,1], [2,3,1], [2,1,3], [1,2,1,2].

%e T(6,2) = 2: [3,2,1], [2,1,2,1].

%e T(7,0) = 5: [7], [3,4], [1,6], [2,5], [1,2,4].

%e T(7,1) = 13: [4,3], [6,1], [5,2], [2,1,4], [4,1,2], [1,4,2], [2,3,2], [3,1,3], [1,5,1], [2,4,1], [1,2,3,1], [1,3,1,2], [1,2,1,3].

%e T(7,2) = 5: [4,2,1], [2,1,3,1], [3,1,2,1], [1,3,2,1], [1,2,1,2,1].

%e Triangle T(n,k) begins:

%e 00:   1;

%e 01:   1;

%e 02:   1;

%e 03:   2,   1;

%e 04:   2,   2;

%e 05:   3,   4;

%e 06:   4,   8,   2;

%e 07:   5,  13,   5;

%e 08:   6,  21,  12;

%e 09:   8,  33,  27,   3;

%e 10:  10,  50,  53,  11;

%e 11:  12,  73,  98,  31;

%e 12:  15, 106, 174,  78,   5;

%e 13:  18, 150, 296, 175,  22;

%e 14:  22, 209, 486, 363,  72;

%e 15:  27, 289, 781, 715, 204, 8;

%p b:= proc(n, i) option remember; `if`(n=0, 1, expand(

%p       add(`if`(j=i, 0, b(n-j, j)*`if`(j<i, x, 1)), j=1..n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

%p seq(T(n), n=0..20);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[If[j == i, 0, b[n-j, j]*If[j<i, x, 1]], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Feb 13 2015, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000009, A241691, A241692, A241693, A241694, A241695, A241696, A241697, A241698, A241699, A241700.

%Y Row sums give A003242.

%Y T(3n,n) = A000045(n+1).

%Y T(3n+1,n) = A129715(n) for n>0.

%Y Cf. A238344, A285994.

%K nonn,tabf

%O 0,4

%A _Alois P. Heinz_, Apr 27 2014